MATHEMATICS 150

(2000-2001)

PROBLEM SET 3

Solutions are due on Monday, November 20. Solutions may be submitted in the lecture or may be delivered by 4:00 pm to the instructor's office (CC F30). Print your name on the upper right-hand corner of the front page.

1.	In order to provide some housing relief in a municipality, it is proposed to change the zoning of a parcel of land near a school and playground to allow for high-density residential development and some shopping facilities instead of low-density residential development. To obtain an initial impression of public reaction to the proposal, including possible differences based on how close people lived to the land in question, a news team conducted a poll in a shopping mall close to the area. Respondents were selected at random and asked about their degree of support or opposition. (They were coaxed into indicating some reaction other than "neutral".) As well, on the basis of a map, respondents were classified according to how close they lived to the area. The data are stored in a file HOUSE.DAT which is available at

http://www.trentu.ca/academic/math/courses/150/files/house.dat

Each line of the file contains a respondent's area SAME, ADJACENT or OTHER, left justified in columns 2 through 9, and the respondent's degree of support/opposition STR_SUP, MILD_SUP, MILD_OPP, or STR_OPP, left justified in columns 22 through 29.

a)	Enter these data into Minitab or other software of your choice (You must use an appropriate FORMAT in MINITAB to enter the data.) and use the software to display these data with a cross-tabulation display with appropriate detail. Arrange the display so that the categories are in a "natural" order and so that responses are in rows and the areas are in columns.

b)	Illustrate the information in the cross-tabulation display with an appropriate diagram. (It is not necessary to use software for this diagram.)

2.	In a study on different procedures for accessing computer files, fifteen sample requests were considered for one process. For each request, the time to access the requested file was noted and a measure of the system load in effect at the time of the request was measured as well. It was of interest to predict the time to access a file with this process for varying values of the system load. For the following fifteen pairs of sample values, x represents a measure of the system load and y represents the time to access the requested file.

x	49.8	71.2	38.1	43.5	58.2	54.2	15.7	63.1

y	2.4	6.0	1.9	2.5	3.3	3.0	1.7	3.6

x	50.6	62.9	53.4	46.2	25.0	49.6	82.6

y	3.2	4.3	2.9	2.2	1.6	2.8	7.0

a)	Plot a scatter diagram of y vs x.

b)	Does the relationship appear to be linear for the range sampled?

c)	Calculate the correlation coefficient for x and y.

d)	Determine the least squares regression line for y vs x.

e)	Predict times for file requests in similar conditions with system load values of x = 30, 50 and 70.

f)	Plot the regression line on a scatter diagram of the data.

g)	Comment on the appropriateness (in terms of accuracy and reality) of predictions obtained by extrapolating beyond the sampling range.

3.	Suppose that, for the study in Problem 2, it is decided to use the least squares procedure with transformed data to produce a fitted equation

y = aebx

a)	What are xnew and ynew in terms of xold and yold? What are b0 and b1in terms of a and b?

b)	What transformation, if any, must be used to change the new fitted values to fitted values for the original data?

c)	Using the sample data, determine the linear regression line for ynew vs xnew and plot this fitted line on a scatter diagram of the transformed data.

d)	What is the correlation between ynew and xnew?

e)	Determine the fitted curve for the original data that results from part c) and plot it on a scatter diagram of the original data.

f)	What extrapolation problem(s) would this model have, if any?

4.	It is assumed that package contents will experience shrinkage due to moisture loss, etc. during storage. This assumption leads to a model y = b1 x, where x is the initial weight and y is the weight after storage. The coefficient b1 represents the proportion of the contents remaining after storage. (If there is a 7% loss through shrinkage, then 93% will remain and b1 will be 0.93.) A sample of 12 packages is to be used to estimate the loss during a set storage time. The sample packages were weighed, then stored for the length of time under study, then weighed again. The resulting weights were:

initial weight x	122.7	124.0	131.1	122.8	126.1	123.1	127.7	120.7	126.5	122.1	124.7	128.3

final weight y	116.4	118.8	127.3	116.4	120.9	117.2	121.0	113.7	119.2	116.0	117.7	122.4

a)	Sketch a scatter diagram of y vs x.

b)	Determine the least squares regression line to predict y from x on the basis of the assumed model.

c)	What type of regression is used in part b)?

d)	Plot the fitted line on a scatter diagram of the data.

e)	What is the estimated percentage loss from shrinkage?

f)	Predict after-storage weight for a package with an initial weight of 125.